Multiple-valued logics for theorem-proving in first order logic with equality

In this paper, we present a system that translates first order logic (FOL) formulas into natural language (NL) sentences. The motivation comes from an intelligent tutoring system teaching logic as a knowledge representation language, where it is used as a means for feedback to the users. FOL to NL conversion is achieved by using a rule-based approach, where we exploit the pattern matching capabilities of rules. So, the system consists of a rule-based component and a lexicon. The rule-based unit implements the conversion process, which is based on a linguistic analysis of a FOL sentence, and the lexicon provides lexical and grammatical information that helps in producing the NL sentences. The whole system is implemented in Jess, a java-based expert system shell. The conversion process currently covers a restricted set of FOL formulas.

Approximate pattern matching and transitive closure logics

The role of games in computing science has been given some consideration in the literature. We aim to design games that can be used for teaching logic. Specifically, we build on the Minefield game used in the Stanford Intro to Logic course, and present a version, which we call MineFOL, that incorporates time constraints. We also move beyond the Stanford examples in that we provide a formalisation of the game, with a view to eventually generating game instances automatically as well as from user input, and show how MineFOL can be used as a learning tool. We also developed an online platform supporting learners in at least three ways. First, the learners can practice reasoning in First Order Logic (FOL) and proving strategies such as resolution through various MineFOL games. Second, the learners can practice formalisation in FOL, by allowing learners to build their own games. Third, the learners become aware of the need of interleaving various technologies to solve a logicbased task. In this line, MineFOL is formalised in such way to allow augmenting reasoning in FOL with search strategies.

We study Sowa’s conceptual graphs (CGs) with both existential and universal quantifiers. We explore in detail the existential fragment. We extend and modify Sowa’s original graph derivation system with new rules and prove the soundness and completeness theorem with respect to Sowa’s standard interpretation of CGs into first order logic (FOL). The proof is obtained by reducing the graph derivation to a question-answering problem. The graph derivation can be equivalently obtained by querying a Definite Horn Clauses program by a conjunction of positive atoms. Moreover, the proof provides an algorithm for graph derivation in a pure proof-theoretic fashion, namely by means of a slight enhancement of the standard PROLOG interpreter. The graph derivation can be rebuilt step-by-step and constructively from the resolution-based proof. We provide a notion of CGs in normal form (the table of the conceptual graph) and show that the PROLOG interpreter also gives a projection algorithm between normal CGs. The normal forms are obtained by extending the FOL language by witnesses (new constants) and extending the graph derivation system. By applying iteratively a set of rules the reduction process terminates with the normal form of a conceptual graph. We also show that graph derivation can be reduced to a question-answering problem in propositional datalog for a subclass of simple CGs. The embedding into propositional datalog makes the complexity of the derivation polynomial.

The expressiveness of First Order Logic (FOL) languages is severely counterbalanced by the complexity of matching formulas on a universe. Matching is an instance of the class of Constraint Satisfaction Problems (CSP), which have shown to undergo a phase transition with respect to two order parameters: constraint density and constraint tightness. This paper analyzes the problem of satisfying FOL Horn clauses in the light of these recent results. By means of an extensive experimental analysis, we show how Horn clause verification exhibits a typical phase transition with respect to the number of binary (or greater arity) predicates, and with respect to the ratio between the number of constants in the universe and the cardinality of the basic predicates extension.

Software Product Line (SPL) is a new methodology that develops software products by configuring a software product from artifacts repository. Variability is one of the important issues in designing SPL. It reflects the diversity and commonality of the artifacts in a product line. The success of SPL is basically dependent on model's variability. SPL contains three main issues: variability modeling, configuration of new software and the analysis of SPL. Validation is a vital operation among all these issues. In the literature, many methods of modeling variability are proposed but none of them focuses on the validation. In this paper, an intelligent method for validating SPL is introduced. The proposed method is based on two layers. The higher layer is a graphical representation (satisfying visualization condition), and the lower layer is logical representation of the variability using first order logic (FOL). The way that the proposed method can be used to model variability, and support the analysis and validation of SPL is described. Later a new operation in SPL validation issue is presented. Finally, the implementation of the two basic operations in the analysis of SPL is illustrated.

Software product Line (SPL) is an emerging methodology for developing software products. A successful software product is highly dependent on the validity of a SPL. Therefore, validation is a significant process within SPL. In this paper, inconsistency detection is investigated as operation for validating SPL. Intelligent rules are formulated detecting inconsistency based on deducing the results from predefined cases. First, variability is modeled using First Order Logic (FOL) predicates as a prerequisite for inconsistency detection. Later, inconsistency is categorized in three groups. For each group a general form is formulated that can coffer all possible cases. Finally, an intelligent rule (based on FOL) is illustrated for implementing each possibility. As results, all cases of inconsistency in the domain-engineering process are defined.

In this paper we will present a graph-transformation based method for the verification of heterogeneous first order logic (FOL) and Euler/Venn proofs. In previous work, it has been shown that a special collection of directed acyclic graphs (DAGs) can be used interchangeably with Euler/Venn diagrams in reasoning processes. Thus, proofs which include Euler/Venn diagrams can be thought of as proofs with DAGs where steps involving only Euler/Venn diagrams can be treated as particular DAG transformations. Here we will show how the characterization of these manipulations can be used to verify Euler/Venn proofs. Also, a method for verifying the use of heterogeneous Euler/Venn and FOL reasoning rules will be presented that is also based upon DAG transformations .

Logical equivalence between logic programs that are firstorder logic formulas holds between few logic programs, partly because first-order logic does not allow auxiliary programs and data structures to be hidden. As a result of not having such abstractions, logical equivalence will force these auxiliaries to be present in any equivalence program. Higher-order quantification can be use to hide predicates and function symbols. If such higher-order quantification is restricted so that operationally, only hiding is specified, then the cost of such higher-order quantifiers within proof search can be small: one only needs to deal with adding new eigenvariables and clauses involving such eigenvariables. On the other hand, the specification of hiding via quantification can allow for novel and interesting proofs of logical equivalence between programs. This paper will present several example of how reasoning directly on a logic program can benefit significantly if higher-order quantification is used to provide abstractions.KeywordsLogic ProgramLogic ProgrammingSecurity ProtocolLinear LogicHorn ClauseThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we will present a definability theorem for first order logic.<br />This theorem is very easy to state, and its proof only uses elementary tools. To explain the theorem, let us first observe that if M is a model of a theory T in a language L, then, clearly, any definable subset S M (i.e., a subset S = {a | M |= phi(a)} defined by some formula phi) is invariant under all<br />automorphisms of M. The same is of course true for subsets of M" defined<br />by formulas with n free variables.<br /> Our theorem states that, if one allows Boolean valued models, the converse holds. More precisely, for any theory T we will construct a Boolean valued model M, in which precisely the T-provable formulas hold, and in which every (Boolean valued) subset which is invariant under all automorphisms of M is definable by a formula of L.<br />Our presentation is entirely selfcontained, and only requires familiarity<br />with the most elementary properties of model theory. In particular, we have added a first section in which we review the basic definitions concerning<br />Boolean valued models.<br />The Boolean algebra used in the construction of the model will be presented concretely as the algebra of closed and open subsets of a topological space X naturally associated with the theory T. The construction of this space is closely related to the one in [1]. In fact, one of the results in that paper could be interpreted as a definability theorem for infinitary logic, using topological rather than Boolean valued models.

Tractable reasoning via approximation

Our theorem proving system named TTP (which is short for Teaching Theorem Proving) can construct the natural deduction proofs of the first-order logic formulas with quantifiers. It is written in CLISP. It has 4 learning models. Students can use the system to do their logic homework step by step and learn the skills of proving logic theorems. If they don't know how to prove the theorems, they can ask for hints. In our university about 300 students used the system to learn logic and do homework during 3 years, they did 2000 problems in logic.

We show that, assuming the consistency of a supercompact cardinal, the first (weakly) inaccessible cardinal can satisfy a strong form of a Löwenheim–Skolem–Tarski theorem for the equicardinality logic L(I), a logic introduced in [5] strictly between first order logic and second order logic. On the other hand we show that in the light of present day inner model technology, nothing short of a supercompact cardinal suffices for this result. In particular, we show that the Löwenheim–Skolem–Tarski theorem for the equicardinality logic at κ implies the Singular Cardinals Hypothesis above κ as well as Projective Determinacy.

The paper investigates interpretations of propositional and first-order logic in which validity is defined in terms of partial indices; sometimes called possibilities but here understood as non-empty subsets of a set W of possible worlds. Truth at a set of worlds is understood to be truth at every world in the set. If all subsets of W are permitted the logic so determined is classical first-order predicate logic. Restricting allowable subsets and then imposing certain closure conditions provides a modelling for intuitionistic predicate logic. The same semantic interpretation rules are used in both logics for all the operators.

Provenance is critical to the scientific integrity of archived data. It establishes the authenticity of science data by providing a detailed account of its origin, ownership, processing history, and any transformations that occurred over its lifecycle. In addition to validating authenticity, provenance facilitates the scientific evaluation of data and can be decisive in legal contexts regarding ownership and access.The Planetary Data System Information Model (PDS4 IM) provides some support for provenance such as limited processing history and source product identifiers.Nevertheless, the need for a more comprehensive and formalized approach to provenance tracking has been expressed across the PDS for derived data products, superseded unique identifiers, and the curation of data collections over time.To address these needs the W3C PROV Data Model (PROV-DM), Figure 1, has been adopted as the foundation for a robust, interoperable provenance model in PDS4.The PROV-DM Standard PROV-DM is a W3C Recommendation designed to support the interchange of provenance information on the Web. It introduces a generic yet expressive vocabulary for modeling entities, activities, and agents, as well as their interrelations such as wasDerivedFrom, wasGeneratedBy, wasAssociatedWith, and used. The adoption of PROV-DM in PDS4 provides a well-supported foundation for provenance interoperability. Figure 1 - PROV-DM: The W3C PROV Data ModelUsing GPT_4o, phase one consisted of the development of a general model aligned with PROV-DM and implemented as a PDS4 Local Data Dictionary (LDD), an extension to the PDS4 Information Model. As a general model, this LDD can in turn be specialized to support a diverse set of use cases across the PDS archive.Four use cases are currently under consideration. Each is conceptualized below by defining their PROV-DM entities and relationships.Voyager Corrected Image - This historical use case illustrates derivation of a corrected image from raw data and calibration files.Entities: CorrectedProduct, Raw Product, Calibration File Activities: Radiometric/Geometric Correction and Artifact Removal, Voyager Data Acquisition, Processing pipeline “vlev1” Agents: The software module components of “vlev1”, for example, findrx, remrx, and voycal. Relations: wasDerivedFrom, used, wasGeneratedBy, wasAttributedTo, wasAssociatedWith Superseded Logical Identifier (LID) - This use case tracks the relationship between a superseded identifier and a new identifierEntities: Superseded LID, New LID Activities: Superseding Activity Agents: PDS Discipline Node, data engineer Relations: wasDerivedFrom, wasAssociatedWith PDS Collection Stewardship – This use case describes the lifecycle of a collection across multiple versions and curatorial responsibility transfers.Entities: PDS Collection Version 1, PDS Collection Version 2 Activities: Archive, Management Agents: Discipline Node A, Discipline Node B Relations: wasGeneratedBy, wasAssociatedWith, used Planetary Science Observation – This use case captures the entities and relationships involved in a science observation.Investigation Goal: Volatile Mapping on Ryugu Spacecraft: Hayabusa2 (operated by JAXA) Instruments: Near-Infrared Spectrometer (NIRS3), Thermal Infrared Imager (TIR) Activity: Observation Campaign 1 Entities: Digital objects (e.g., spectra, images, thermographs) Targets: Crater A, Boulder Cluster B, Equatorial Ridge, Polar Region, Landing Site Z Relations: wasDerivedFrom, wasGeneratedBy, used, wasAssociatedWith, wasInformedBy A pseudo-XML representation representing some of the entities, activities, agents, and relationships for the first use case is provided below. Voyager Corrected Image Derived From EDR Image The VoyagerISSCorrectedImage was derived from the EDRImage. entity_voyager_corrected_image entity_voyager_edr_image activity_data_correction Some challenges remain in finalizing the implementation strategy:Identify and define the possible reasons for identifier supersession: Superseded, Replaced, Error, etc. Determine the suitability and granularity of W3C PROV relationships for the PDS. Determine appropriate use of bidirectional vs. unidirectional references between data products. Develop policies and procedures for governance to ensure consistent provenance recording and utilization. Conclusion The application of the W3C PROV-DM standard to the PDS4 information model represents a significant step toward improving the traceability, interoperability, and scientific trustworthiness of planetary data. The use cases presented demonstrate a broad applicability across contexts, from spacecraft imaging pipelines to multi-node stewardship models. Continued refinement of relationship types and implementation strategies will ensure alignment with both scientific practice and digital preservation principles.References W3C. (2013). PROV-DM: The PROV Data Model. https://www.w3.org/TR/prov-dm/AcknowledgementsThe research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.During the preparation of this work the author(s) used GPT4o in order to transform natural language to First Order Logic (FOL) and then to several Data Definition Languages (DDL). After using this tool/service, the author(s) reviewed and edited the content as needed and take(s) full responsibility for the content of the publication.© 2025 California Institute of Technology. Government sponsorship acknowledged.